Chaos in the Hyperbolic Restricted 3-Body Problem

Let m 1 = m 2 = 1 be two mass points moving under Newton’s law of attraction in hyperbolic orbits in the xy-plane while their center of mass is fixed at the origin of coordinates. As usual these two masses are called primaries. We suppose that the hyperbolic orbit is a non-collision orbit; that is, the motion of m 1 and m 2 is not on a straight line, or equivalently, their angular momentum is different from zero. We consider a third mass point with infinitely small mass moving on the z-axis (see Figure 1). Since m 3 = 0 the motion of the first two mass points is not affected by the third and from the symmetry of the motion it is clear that the third mass point will remain on the z-axis. The problem is to describe the motion of the infinitely small mass, and then we have a restricted three-body problem that we can the hyperbolic restricted problem. The equation of motion of this problem in the phase space (z, ż, t) is